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This means that, since in the mirror (IIA) picture the ti form the scalar components of the vector multiplets which should span a special K¨ ahler manifold, the ti moduli should also span a special K¨ ahler submanifold within the overall quaternionic manifold spanned by the full hypermultiplets. 26). 33) which is again derived from a prepotential F km which in this case can be evaluated explicitly F km = − 1 Kijk X i X j X k , 3! 35) and we have introduced the homogeneous co-ordinates X I = (1, ti ) to keep the analogy with the special K¨ ahler geometry.

8) XA X0 = (1, z a ). Then these co-ordinates are a good co-ordinate basis on the manifold so that the bottom half of the holomorphic section composed of the periods FA is actually a function of the top-half periods FA (X) and can be generated from a single holomorphic function of homogeneous degree two called the prepotential FA (X) = ∂ F(X) . 10) FAB = ∂A ∂B F . 11) where The bundle H clearly plays an important role in the geometry of special K¨ ahler man- ifolds and in particular its section Ωsk .

16) ∇a¯ Ωsk = ∂a¯ Ωsk = 0 . 17) This gives the covariant derivative, with respect to K¨ ahler transformations, acting on Ωsk . Supersymmetry implies that the vectors in the vector multiplet sector should follow the same geometry as the scalars and indeed this is the case. In fact this relation between the vectors and the scalars is the important constraint through which the whole geometric structure is derived. A particularly important concept is electric-magnetic duality which corresponds to the symplectic symmetry present in the scalar sector.

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Aspects of Moduli Stabilization in String and M-Theory [thesis] by E. Palti

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